Transposition (1 3) and 5-cycle (1 2 3 4 5) generate S5 while transposition (1 3) and 4-cycle (1 2 3 4 4) does not generate S4.
generating-the-symmetric-group-with-a-transposition-and-a-maximal-length-cycle-image
Transposition (1 3) and 5-cycle (1 2 3 4 5) generate S5 while transposition (1 3) and 4-cycle (1 2 3 4 4) does not generate S4.